Solving Rational Equations and Applications

 

Learning Objective(s)

·         Solve rational equations.

·         Check for extraneous solutions.

·         Solve application problems involving rational equations.

 

 

Equations that contain rational expressions are called rational equations. For example,  is a rational equation.

 

You can solve these equations using the techniques for performing operations with rational expressions and the procedures for solving algebraic equations. Rational equations can be useful for representing real-life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time relationships and for modeling work problems that involve more than one person.

 

One method for solving rational equations is to rewrite the rational expressions in terms of a common denominator. Then, since you know the numerators are equal, you can solve for the variable. To illustrate this, let’s look at a very simple equation.

 

 

Since the denominator of each expression is the same, the numerators must be equivalent. This means that x = 2.

 

This is true for rational equations with polynomials too.

 

 

Since the denominators of each rational expression are the same, x + 4, the numerators must be equivalent for the equation to be true. So, x – 5 = 11 and x = 16.

 

Just as with other algebraic equations, you can check your solution in the original rational equation by substituting the value for the variable back into the equation and simplifying.

 

 

 

When the terms in a rational equation have unlike denominators, solving the equation will involve some extra steps. One way of solving rational equations with unlike denominators is to multiply both sides of the equation by the least common multiple of the denominators of all the fractions contained in the equation. This eliminates the denominators and turns the rational equation into a polynomial equation. Here’s an example.

 

 

 

 

Another way to solve a rational equation with unlike denominators is to rewrite each term with a common denominator and then just create an equation from the numerators. This works because if the denominators are the same, the numerators must be equal. The next example shows this approach with the same equation you just solved:

 

 

 

 

In some instances, you’ll need to take some additional steps in finding a common denominator. Consider the example below, which illustrates using what you know about denominators to rewrite one of the expressions in the equation.

 

 

 

 

You could also solve this problem by multiplying each term in the equation by 3 to eliminate the fractions altogether. Here is how it would look.

 

 

 

 

Some rational expressions have a variable in the denominator. When this is the case, there is an extra step in solving them. Since division by 0 is undefined, you must exclude values of the variable that would result in a denominator of 0. These values are called excluded values. Let’s look at an example.

 

 

 

 

Give the excluded values for . Do not solve.   A) B) 2 C) −2, 2 D) −2, 2, 4   Show/Hide Answer A) Incorrect. Excluded values are those values of the variable which result in a 0 in denominator, not in the numerator. The correct answer is −2, 2.   B) 2 Incorrect. 2 is an excluded value, but −2 also results in a 0 in the denominator. The correct answer is −2, 2.   C) −2, 2 Correct. −2 and 2, when substituted into the equation, result in a 0 in the denominator. Since division by 0 is undefined, both of these values are excluded from the solution.   D) −2, 2, 4 Incorrect. While −2 and 2 are excluded, 4 is not excluded because it does not cause the denominator to be 0. The correct answer is −2, 2.

 

 

Let’s look at an example with a more complicated denominator.

 

 

 

 

Solve the equation , m  0 or 2   A) m = 2 B) no solution C) m = 8   Show/Hide Answer A) m = 2 Incorrect. You probably found the common denominator correctly, but forgot to distribute when you were simplifying. You also forgot to check your solution or note the excluded values; m ≠ 2 because it makes the expression on the right side undefined. Multiplying both sides by the common denominator gives , so . The correct answer is m = 8.   B) no solution Incorrect. , so . The solution, 8, is not an excluded value. The correct answer is m = 8.   C) m = 8 Correct. Multiplying both sides of the equation by the common denominator gives , so . The correct answer is m = 8.

 

 

You’ve seen that there is more than one way to solve rational equations. Because both of these techniques manipulate and rewrite terms, sometimes they can produce solutions that don’t work in the original form of the equation. These types of answers are called extraneous solutions. That's why it is always important to check all solutions in the original equations—you may find that they yield untrue statements or produce undefined expressions.

 

 

 

 

 

A “work problem” is an example of a real life situation that can be modeled and solved using a rational equation. Work problems often ask you to calculate how long it will take different people working at different speeds to finish a task. The algebraic models of such situations often involve rational equations derived from the work formula, W = rt. (Notice that the work formula is very similar to the relationship between distance, rate, and time, or d = rt.) The amount of work done (W) is the product of the rate of work (r) and the time spent working (t). The work formula has 3 versions.

 

W = rt

 

 

 

Some work problems include multiple machines or people working on a project together for the same amount of time but at different rates. In that case, you can add their individual work rates together to get a total work rate. Let’s look at an example.

 

 

 

 

Other work problems go the other way. You can calculate how long it will take one person to do a job alone when you know how long it takes people working together to complete the job.

 

 

 

 

As shown above, many work problems can be represented by the equation , where t is the time to do the job together, a is the time it takes person A to do the job, and b is the time it takes person B to do the job. The 1 refers to the total work done—in this case, the work was to paint 1 house.

 

The key idea here is to figure out each worker’s individual rate of work. Then, once those rates are identified, add them together, multiply by the time t, set it equal to the amount of work done, and solve the rational equation.

 

 

Mari and Liam can each wash a car and vacuum its interior in 2 hours. Zach needs 3 hours to do this same job alone. If Zach, Liam, and Mari work together, how long will it take them to clean a car?   A) 20 minutes B) 45 minutes C) 1.2 hours D) 1 hour   Show/Hide Answer A) 20 minutes Incorrect. It looks like you divided one hour by 3 to arrive at 20 minutes. Remember that Zach is working at a different rate than Mari and Liam, so you cannot do straight division. The correct answer is 45 minutes.   B) 45 minutes Correct. According to the formula, . Mari and Liam each have a rate of  car in one hour, and Zach’s rate is  car in one hour. Working together, they have a rate of , or . W is one car, so the formula becomes  = . This means , so , and t = . It takes three-quarters of an hour, or 45 minutes, to clean one car.   C) 1.2 hours Incorrect. Mari and Liam clean a car in two hours each, not as a team. They each have a rate of  car in one hour, and Zach’s rate is  car in one hour. Working together, they have a rate of . The correct answer is 45 minutes.   D) 1 hour Incorrect. This is the time it would take for Mari and Liam to clean one car together. Since Zach is helping, it will take less time than that. The correct answer is 45 minutes.

 

 

You can solve rational equations by finding a common denominator. By rewriting the equation so that all terms have the common denominator, you can solve for the variable using just the numerators. Or, you can multiply both sides of the equation by the least common multiple of the denominators so that all terms become polynomials instead of rational expressions. Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule.

 

An important step in solving rational equations is to reject any extraneous solutions from the final answer. Extraneous solutions are solutions that don't satisfy the original form of the equation because they produce untrue statements or are excluded values that make a denominator equal to 0.